Explanation of noise texture node settings in 2.8
You can better understand texture space using a color grid image (2D), with "generated" coordinates, which is the default for noise textures.

You have to imagine the 2D texture laying on the floor of the bounding box of your object (the tiniest box that can contain all your object), with lower left pixel as starting and pivot point, which will stay in the lower left and front corner of the box.

In this case all points on the surface which share the same X and Y coordinates will get the same colour, as you can see on the front face of Suzanne, unlike its top, where the texture is projected nearly undistorted because every point has different X and Y coordinates.

The same happens with noise texture when set to 2D.

If set to 3D, points on surface with different Z coordinates will have a different random color even if they share X and Y; it works as "voxels" instead of pixels (and is the "classic" use).

1D means that a random color is outputted, based on the W parameter value only.

4D is like 3D, but you can vary the W parameter to have a different random combination (similar to what happens when you animate one coordinate parameter).

The more dimensions used, the more calculating power is needed, so use the minimum dimension you need.